L(s) = 1 | − 0.537i·2-s − 1.73·3-s + 3.71·4-s − 0.803·5-s + 0.931i·6-s + 5.11·7-s − 4.14i·8-s + 2.99·9-s + 0.431i·10-s + 17.7i·11-s − 6.42·12-s − 24.6i·13-s − 2.74i·14-s + 1.39·15-s + 12.6·16-s + 18.2·17-s + ⋯ |
L(s) = 1 | − 0.268i·2-s − 0.577·3-s + 0.927·4-s − 0.160·5-s + 0.155i·6-s + 0.730·7-s − 0.518i·8-s + 0.333·9-s + 0.0431i·10-s + 1.61i·11-s − 0.535·12-s − 1.89i·13-s − 0.196i·14-s + 0.0927·15-s + 0.788·16-s + 1.07·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.904 + 0.425i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.904 + 0.425i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.61165 - 0.360069i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.61165 - 0.360069i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + 1.73T \) |
| 59 | \( 1 + (53.3 + 25.1i)T \) |
good | 2 | \( 1 + 0.537iT - 4T^{2} \) |
| 5 | \( 1 + 0.803T + 25T^{2} \) |
| 7 | \( 1 - 5.11T + 49T^{2} \) |
| 11 | \( 1 - 17.7iT - 121T^{2} \) |
| 13 | \( 1 + 24.6iT - 169T^{2} \) |
| 17 | \( 1 - 18.2T + 289T^{2} \) |
| 19 | \( 1 - 28.5T + 361T^{2} \) |
| 23 | \( 1 + 11.8iT - 529T^{2} \) |
| 29 | \( 1 - 9.01T + 841T^{2} \) |
| 31 | \( 1 + 4.94iT - 961T^{2} \) |
| 37 | \( 1 - 39.7iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 38.0T + 1.68e3T^{2} \) |
| 43 | \( 1 + 19.2iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 65.6iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 40.1T + 2.80e3T^{2} \) |
| 61 | \( 1 - 110. iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 30.7iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 95.0T + 5.04e3T^{2} \) |
| 73 | \( 1 - 71.2iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 13.3T + 6.24e3T^{2} \) |
| 83 | \( 1 + 142. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 128. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 97.1iT - 9.40e3T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.13356196427521252157611877632, −11.59653208706668374889451381934, −10.34384097897141084342002711052, −9.937007287712751980073594287226, −7.84383269126845759568953662827, −7.40466907510709219626830340166, −5.90899474051703746998813374993, −4.86763672207425459506311760115, −3.09269846904502707001960919090, −1.38030843357097825901642573443,
1.52834207301522552586724285155, 3.49225204699854748498671316408, 5.23283628239917742397739038836, 6.15075266629380534092813964451, 7.27172727887612883245324189386, 8.201047591896286582999821476067, 9.568261286597801914033562858337, 10.93485037320425087682255257650, 11.60422634462394494579203282345, 11.97859003141396509261485501326